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Inventory Control. Chapter 15. OBJECTIVES . Inventory System Defined Inventory Costs Independent vs. Dependent Demand Single-Period Inventory Model Multi-Period Inventory Models: Basic Fixed-Order Quantity Models Multi-Period Inventory Models: Basic Fixed-Time Period Model

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  • Inventory System Defined
  • Inventory Costs
  • Independent vs. Dependent Demand
  • Single-Period Inventory Model
  • Multi-Period Inventory Models: Basic Fixed-Order Quantity Models
  • Multi-Period Inventory Models: Basic Fixed-Time Period Model
  • Miscellaneous Systems and Issues
inventory system
Inventory System
  • Inventory is the stock of any item or resource used in an organization and can include: raw materials, finished products, component parts, supplies, and work-in-process
  • An inventory system is the set of policies and controls that monitor levels of inventory and determines what levels should be maintained, when stock should be replenished, and how large orders should be
purposes of inventory
Purposes of Inventory

1. To maintain independence of operations

2. To meet variation in product demand

3. To allow flexibility in production scheduling

4. To provide a safeguard for variation in raw material delivery time

5. To take advantage of economic purchase-order size

inventory costs
Inventory Costs
  • Holding (or carrying) costs
    • Costs for storage, handling, insurance, etc
  • Setup (or production change) costs
    • Costs for arranging specific equipment setups, etc
  • Ordering costs
    • Costs of someone placing an order, etc
  • Shortage costs
    • Costs of canceling an order, etc
independent vs dependent demand

Independent Demand (Demand for the final end-product or demand not related to other items)

Dependent Demand

(Derived demand items for component parts,


raw materials, etc)

Independent vs. Dependent Demand




Component parts

inventory systems
Inventory Systems
  • Single-Period Inventory Model
    • One time purchasing decision (Example: vendor selling t-shirts at a football game)
    • Seeks to balance the costs of inventory overstock and under stock
  • Multi-Period Inventory Models
    • Fixed-Order Quantity Models
      • Event triggered (Example: running out of stock)
    • Fixed-Time Period Models
      • Time triggered (Example: Monthly sales call by sales representative)
single period inventory model
Single-Period Inventory Model

This model states that we should continue to increase the size of the inventory so long as the probability of selling the last unit added is equal to or greater than the ratio of: Cu/Co+Cu

single period model example
Single Period Model Example
  • Our college basketball team is playing in a tournament game this weekend. Based on our past experience we sell on average 2,400 shirts with a standard deviation of 350. We make $10 on every shirt we sell at the game, but lose $5 on every shirt not sold. How many shirts should we make for the game?

Cu = $10 and Co= $5; P≤ $10 / ($10 + $5) = .667

Z.667 = .432 (use NORMSDIST(.667) or Appendix E)

therefore we need 2,400 + .432(350) = 2,551 shirts

multi period models fixed order quantity model model assumptions part 1
Multi-Period Models:Fixed-Order Quantity Model Model Assumptions (Part 1)
  • Demand for the product is constant and uniform throughout the period
  • Lead time (time from ordering to receipt) is constant
  • Price per unit of product is constant
multi period models fixed order quantity model model assumptions part 2
Multi-Period Models:Fixed-Order Quantity Model Model Assumptions (Part 2)
  • Inventory holding cost is based on average inventory
  • Ordering or setup costs are constant
  • All demands for the product will be satisfied (No back orders are allowed)
basic fixed order quantity model and reorder point behavior

4. The cycle then repeats.

1. You receive an order quantity Q.


of units

on hand







2. Your start using them up over time.

3. When you reach down to a level of inventory of R, you place your next Q sized order.


R = Reorder point

Q = Economic order quantity

L = Lead time

Basic Fixed-Order Quantity Model and Reorder Point Behavior
cost minimization goal

Total Cost

Annual Cost of

Items (DC)


Cost Minimization Goal

By adding the item, holding, and ordering costs together, we determine the total cost curve, which in turn is used to find the Qopt inventory order point that minimizes total costs







Ordering Costs

Order Quantity (Q)

basic fixed order quantity eoq model formula
Basic Fixed-Order Quantity (EOQ) Model Formula

TC=Total annual cost

D =Demand

C =Cost per unit

Q =Order quantity

S =Cost of placing an order or setup cost

R =Reorder point

L =Lead time

H=Annual holding and storage cost per unit of inventory


Annual =













deriving the eoq
Deriving the EOQ

Using calculus, we take the first derivative of the total cost function with respect to Q, and set the derivative (slope) equal to zero, solving for the optimized (cost minimized) value of Qopt

We also need a reorder point to tell us when to place an order

eoq example 1 problem data
EOQ Example (1) Problem Data

Given the information below, what are the EOQ and reorder point?

Annual Demand = 1,000 units

Days per year considered in average

daily demand = 365

Cost to place an order = $10

Holding cost per unit per year = $2.50

Lead time = 7 days

Cost per unit = $15

eoq example 1 solution
EOQ Example (1) Solution

In summary, you place an optimal order of 90 units. In the course of using the units to meet demand, when you only have 20 units left, place the next order of 90 units.

eoq example 2 problem data
EOQ Example (2) Problem Data

Determine the economic order quantity

and the reorder point given the following…

Annual Demand = 10,000 units

Days per year considered in average daily demand = 365

Cost to place an order = $10

Holding cost per unit per year = 10% of cost per unit

Lead time = 10 days

Cost per unit = $15

eoq example 2 solution
EOQ Example (2) Solution

Place an order for 366 units. When in the course of using the inventory you are left with only 274 units, place the next order of 366 units.

fixed time period model with safety stock formula
Fixed-Time Period Model with Safety Stock Formula

q = Average demand + Safety stock – Inventory currently on hand

multi period models fixed time period model determining the value of s t l
Multi-Period Models: Fixed-Time Period Model: Determining the Value of sT+L
  • The standard deviation of a sequence of random events equals the square root of the sum of the variances
example of the fixed time period model
Example of the Fixed-Time Period Model

Given the information below, how many units should be ordered?

Average daily demand for a product is 20 units. The review period is 30 days, and lead time is 10 days. Management has set a policy of satisfying 96 percent of demand from items in stock. At the beginning of the review period there are 200 units in inventory. The daily demand standard deviation is 4 units.

example of the fixed time period model solution part 1
Example of the Fixed-Time Period Model: Solution (Part 1)

The value for “z” is found by using the Excel NORMSINV function, or as we will do here, using Appendix D. By adding 0.5 to all the values in Appendix D and finding the value in the table that comes closest to the service probability, the “z” value can be read by adding the column heading label to the row label.

So, by adding 0.5 to the value from Appendix D of 0.4599, we have a probability of 0.9599, which is given by a z = 1.75

example of the fixed time period model solution part 2
Example of the Fixed-Time Period Model: Solution (Part 2)

So, to satisfy 96 percent of the demand, you should place an order of 645 units at this review period

price break model formula
Price-Break Model Formula

Based on the same assumptions as the EOQ model, the price-break model has a similar Qopt formula:

i = percentage of unit cost attributed to carrying inventory

C = cost per unit

Since “C” changes for each price-break, the formula above will have to be used with each price-break cost value

price break example problem data part 1
Price-Break Example Problem Data (Part 1)

A company has a chance to reduce their inventory ordering costs by placing larger quantity orders using the price-break order quantity schedule below. What should their optimal order quantity be if this company purchases this single inventory item with an e-mail ordering cost of $4, a carrying cost rate of 2% of the inventory cost of the item, and an annual demand of 10,000 units?

Order Quantity(units)Price/unit($)

0 to 2,499 $1.20

2,500 to 3,999 1.00

4,000 or more .98

price break example solution part 2
Price-Break Example Solution (Part 2)

First, plug data into formula for each price-break value of “C”

Annual Demand (D)= 10,000 units

Cost to place an order (S)= $4

Carrying cost % of total cost (i)= 2%

Cost per unit (C) = $1.20, $1.00, $0.98

Next, determine if the computed Qopt values are feasible or not

Interval from 0 to 2499, the Qopt value is feasible

Interval from 2500-3999, the Qopt value is not feasible

Interval from 4000 & more, the Qopt value is not feasible

price break example solution part 3
Price-Break Example Solution (Part 3)

Since the feasible solution occurred in the first price-break, it means that all the other true Qopt values occur at the beginnings of each price-break interval. Why?

Because the total annual cost function is a “u” shaped function

Total annual costs

So the candidates for the price-breaks are 1826, 2500, and 4000 units

0 1826 2500 4000 Order Quantity

price break example solution part 4
Price-Break Example Solution (Part 4)

Next, we plug the true Qopt values into the total cost annual cost function to determine the total cost under each price-break


= $12,043.82

TC(2500-3999)= $10,041

TC(4000&more)= $9,949.20

Finally, we select the least costly Qopt, which is this problem occurs in the 4000 & more interval. In summary, our optimal order quantity is 4000 units

miscellaneous systems optional replenishment system

q = M - I

Actual Inventory Level, I


Miscellaneous Systems:Optional Replenishment System

Maximum Inventory Level, M


Q = minimum acceptable order quantity

If q > Q, order q, otherwise do not order any.

miscellaneous systems bin systems



One-Bin System

Periodic Check

Miscellaneous Systems:Bin Systems

Two-Bin System

Order One Bin of


Order Enough to

Refill Bin

abc classification system


% of

$ Value






% of




ABC Classification System
  • Items kept in inventory are not of equal importance in terms of:
    • dollars invested
    • profit potential
    • sales or usage volume
    • stock-out penalties

So, identify inventory items based on percentage of total dollar value, where “A” items are roughly top 15 %, “B” items as next 35 %, and the lower 65% are the “C” items

inventory accuracy and cycle counting
Inventory Accuracy and Cycle Counting
  • Inventory accuracy refers to how well the inventory records agree with physical count
  • Cycle Counting is a physical inventory-taking technique in which inventory is counted on a frequent basis rather than once or twice a year